Path Tracing. Michael Doggett Department of Computer Science Lund university Michael Doggett

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1 Path Tracing Michael Doggett Department of Computer Science Lund university 2012 Michael Doggett

2 Outline Light transport notation Radiometry - Measuring light Illumination Rendering Equation Monte Carlo sampling Russian roulette Path tracing Image based lighting 2011 Michael Doggett

3 This is what we want: Courtesy of Henrik Wann Jensen Courtesy of Paul Debevec Images courtesy of Illuminate Labs 2008 Tomas Akenine-Möller!3

4 Isn t Whitted-style ray tracing enough? Does not give truly realistic images Why? Does not solve the entire rendering equation Miss several phenomenon: Caustics Indirect illumination (color bleeding, ambient lighting) Soft shadows This lecture will present Global Illumination techniques: path tracing and image-based lighting. Images courtesy of Henrik Wann Jensen University of California San Diego 2008 Tomas Akenine-Möller!4

5 Notation after Paul Heckbert, SIGGRAPH 90 Light transport notation Useful tool for thinking about global illumination (GI) Follow light paths The endpoints of straight paths can be: L : light source E : the eye S : a specular reflection D: a diffuse reflection G: semi-diffuse (glossy) reflection Regular expressions can be used: (K)+ : one or more of K (K)* : zero or more of K (K)? : zero or one of K (K M) : a K or an M event 2008 Tomas Akenine-Möller!5

6 Examples of light transport notation light B A Specular ball eye diffuse floor and wall!path A: LDDDE!Path B: LSDSDE 2008 Tomas Akenine-Möller!6

7 Light transport notation: why? The ultimate goal is to simulate all light paths: L(S G D)*E Using this notation, we can find what Whitted ray tracing can handle: LDS*E LS*E = LD?S*E Or if we include glossy surfaces: LD?(G S)*E This is clearly not L(S G D)*E! 2008 Tomas Akenine-Möller!7

8 The Rendering Equation It extends the range of optical phenomena which can be effectively simulated. From paper of the same name, The Rendering Equation, by James T. Kajiya, SIGGRAPH Michael Doggett

9 Radiometry Flux (radiant power): [W ] Big Phi The total energy that flows from/to/through a surface Irradiance: E = d da apple W m 2 Incoming radiant power per unit surface area Radiosity (radiant exitance) Same as irradiance but outgoing 2012 Michael Doggett

10 Radiometry Solid Angle, = A r 2 n x Like the 2D version of an angle Shape of grey area doesn t matter Dimensionless expressed in steradians x r A Solid angle of sphere is Michael Doggett

11 x Radiometry Small Theta Radiance: L(x! ) N L L(x ->Θ) 5 dimensions: position (x) and direction vector (Θ) Important: captures appearance of objects Flux per unit projected area (E) per unit solid angle da θ dω Small Omega Solid angle - subtends area on hemisphere! θ Incoming flux is spread out over a larger area with bigger angles, the cosine term compensates for this 2012 Michael Doggett

12 Rendering Equation L is radiance = most important quantity - captures appearance of objects in a scene. - what we store in a pixel. Energy conservation - total outgoing radiance at a point is the sum of the emitted and reflected radiance. Incoming - Big Psi BRDF The equation above is the Rendering Equation : most important equation in graphics It is a Fredholm equation of the second kind: L is both to the left, and inside the integral Section 11.6 in book, but it uses a different notation 2008 Tomas Akenine-Möller!12

13 Rendering Equation - Lr Why not sample light sources separately? They give most of the appearance (usually)! + Direct illumination 2008 Tomas Akenine-Möller Indirect illumination!13

14 Direct Illumination: L direct Some definitions: Is the closest positive intersection along ray that starts at x and has direction Ψ V(x,y) is the visibility function: is 0 if x occluded from y, otherwise 1. Approach: integrate over the light surface area instead of over the hemisphere The geometry term: G(x,y)V(x,y) is often called radiance transfer 2008 Tomas Akenine-Möller!14

15 Indirect Illumination The incoming radiance at x in direction Ψ is the same as the outgoing radiance from y in direction -Ψ the ray-casting operation, finds the closest positive intersection along ray that starts at x and has direction Ψ 2008 Tomas Akenine-Möller!15

16 GI illustrated + Direct illumination Indirect illumination Recursion Illustrated: (dashed lines are shadow rays) 2008 Tomas Akenine-Möller!16

17 L indirect Change from integrating Cartesian coordinate (solid angle) to (hemi-)spherical coordinates: J is the Jacobian of the coordinate transform 2008 Tomas Akenine-Möller!17

18 Monte Carlo Integration Need to integrate incoming light over hemisphere Estimate definite integral using random samples Monte Carlo estimate 2011 Michael Doggett

19 Monte Carlo Recap X is stochastic random variable, drawn from PDF p(x) How to evaluate the integral of f(x) when x is drawn from PDF p(x)? x i drawn from PDF See chapter 10 for more explanation Copyright 2008 Tomas Akenine-Möller!19

20 Practical direct illumination using MC The PDF, p(y) may be uniform, i.e., 1/A! 2008 Tomas Akenine-Möller!20

21 Russian roulette We need to be able to terminate recursion Bias is the error (possibly) introduced with some technique: E[I MC ]-I (mean of MC minus real solution) Avoid incorrect lighting contributions (bias) Light bounces around infinitely... How to avoid infinite path lengths without bias? 2008 Tomas Akenine-Möller!21

22 Russian roulette for indirect illumination That is, we can stop recursion with a probability of α=1-p, where α is the absorption probability If photon not absorbed, diffuse or specular? Random number r from [0,1] If(r < ρ d ) then shoot diffuse ray! - randomly on hemisphere... Else if(r < ρ d +ρ s ) then shoot specular ray Else absorb Use average reflectance E.g., ρ d =(ρ d,red + ρ d,green + ρ d,blue )/3 Or special reflectivity parameter of material As done in programming assignments 2008 Tomas Akenine-Möller!22

23 Practical indirect illumination How to choose random direction on the hemisphere? Uniformly distributed Instead: I.e., choose random θ and ϕ, and then use The sin θ to correct for non-uniform distribution Not efficient! 2008 Tomas Akenine-Möller!23

24 Importance Sampling Choose the PDF proportional to a factor in the equation that we want to integrate i.e. send more rays near the normal For rendering equation, this can be: cos(ψ i,n x ) Included in programming assignment 3 BRDF Incident radiance Combination Tomas Akenine-Möller!24

25 Path tracing algorithm Shoot many rays through each pixel Compute direct illumination Compute indirect illumination (recursive) Use Russian roulette to terminate recursion without bias Use Hemispherical Monte Carlo sampling! Important for path tracing: Shoot one ray to the light sources Shoot one ray over the hemisphere Reason: do not want to spend more rays that are deeper down in the paths 2008 Tomas Akenine-Möller!25

26 Conventional Ray Tracing vs. Path Tracing ray path ordinary surfaces light surfaces figure after The Rendering Equation, Jim T. Kajiya, ACM SIGGRAPH Michael Doggett

27 Path tracing example 8 spp 13 sec 200 spp 5 min 1000 spp 25min spp 10.3 hrs From Kevin Beason s GI in 99 lines of C Michael Doggett

28 Image Based Lighting 2012 Michael Doggett

29 Image-Based Lighting Enables very realistic lighting In a simple way Not only primary light sources illuminate a point Basic idea: Capture an image of the environment Each pixel in this image is treated as a light source During rendering, use these lights! 2008 Tomas Akenine-Möller!29

30 Example images Images courtesy of Paul Debevec 2008 Tomas Akenine-Möller!30

31 More example images Images courtesy of Henrik Wann Jensen 2008 Tomas Akenine-Möller!31

32 Capturing environment maps Many ways: (light probes) Take a photograph of a highly reflective sphere Fish-eye lens: 2 images needed Special device: rotating image sensors Need high dynamic range images! 3 bytes not enough for RGB " 3 floats Why? Because the range of brightness values is larger than [0,255] 2008 Tomas Akenine-Möller!32

33 Example of image with high dynamic range Image courtesy of Paul Debevec Lowest intensity is about 20 Highest intensity is about 12,000 Dynamic range is 1:600! 2008 Tomas Akenine-Möller!33

34 High Dynamic Range Images Dynamic range of image: brightest region/dimmest region Very briefly: A digital camera uses a CCD (charge coupled device) array as a sensor. However, these values are mapped, using a non-linear function in order to display them on screen To recover the entire range of the image, take several photos of the same scene with different exposure times Non-linear function: from an overdetermined system using SVD Use function to recover HDR image (one float per RGB) There are also cameras that can take 12 bits per color component directly For more details: Debevec & Malik, Recovering High Dynamic Range Radiance Maps from Photographs, SIGGRAPH 97, pp Tomas Akenine-Möller!34

35 Example of image with different exposure times Images courtesy of Paul Debevec 2008 Tomas Akenine-Möller!35

36 How to capture an image of the environment? Use simplest method: image of highly reflective sphere We get an image of the entire environment, if we can assume that the camera is infinitely far away (or if the sphere is very small) 2008 Tomas Akenine-Möller!36

37 Examples of environment maps Dynamic Range: 1:200,000 Images courtesy of Paul Debevec 2008 Tomas Akenine-Möller!37

38 How to use environment map Environment map Often need many rays!! May need varying number of rays depending on frequency content of env. map Note: in path tracing, we only shoot one ray at every intersection point, but many rays through each pixel 2008 Tomas Akenine-Möller!38

39 HDR file formats EXR ILM 16-bit float PFM (Portable Float Map) Public open source Binary, but no compression 2012 Michael Doggett

40 IBL: some notes Even simpler to code with a cube map of the environment Transform sphere map into cube map once before rendering Debevec uses angular map (can use sphere map too) Nice tutorial by Paul Debevec, Image-Based Lighting, IEEE Computer Graphics & Applications, March/April 2002, pp Check out light probe gallery + HDR shop How to do it fast? Agarwal, Ramamoorthi, Belongie and Jensen, Structured Importance Sampling of Environment Maps, SIGGRAPH 2003, pp Probably takes weeks to implement though Tomas Akenine-Möller!40

41 Next Friday Lab 2: BVH Next Week Monday Seminar : Path Tracing Lab (Rasmus) Tuesday Lecture : Photon Mapping Guest Lecture, May 12 : Jonas Gustavsson, Sony Production Rendering 2011 Michael Doggett

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